When it comes to handstand endurance there are probably two different things you want to shoot for.
The first is to hold a handstand against the wall for as long as you can.
Without having to balance you can just focus on building your body’s ability to sustain that position. By doing this you’re building endurance that certainly can transfer over to doing freestanding handstands.
Another aspect of this is that you’ll be able to practice more, because each move won’t be as fatiguing as before when you build up your endurance.
The second way is to work on holding a handstand for longer and longer periods of time out in the open. Here’s a rather boring video showing this in action.
Now a handstand of about a minute and twenty seconds is no world record, though it is far better then most people can manage.
I would also caution you if you’re trying to build up your skill and endurance that it is far better to stand in one place. I also came across this video of a handstand endurance competition, but these people are walking all over the place on their hands.